Problem 75 - Elementary Algebra

Question

Simplify \(x^{8} y^{7}\left(\frac{x^{4} y^{8}}{x^{3} y^{4}}\right)\).

Step-by-Step Solution

Answer

Question: Simplify the expression: \((x^{8}y^{7})(\frac{x^4y^8}{x^3y^4})\) Answer: \(x^9y^{11}\)

Step 1: Simplify the fraction

Here, we begin by simplifying the expression \(\frac{x^{4}y^{8}}{x^{3}y^{4}}\). To simplify the fraction, we will apply the property \(\frac{a^m}{a^n} = a^{m-n}\) for both \(x\) and \(y\) terms. \(\frac{x^{4}y^{8}}{x^{3}y^{4}} = x^{4-3}y^{8-4} = x^1y^4 = xy^4\)

Step 2: Multiply the simplified fraction with the first term

Now, we will multiply the simplified fraction \(xy^4\) with the first term \(x^{8}y^{7}\). Again, we will use the property \(a^{m} \cdot a^{n} = a^{m+n}\) to multiply both terms. \((x^{8}y^{7})(xy^4) = x^{8}x\cdot y^{7}y^4 = x^{8+1}y^{7+4} = x^9y^{11}\) The simplified expression is \(x^9y^{11}\).

Key Concepts

Simplifying Expressions

When it comes to simplifying expressions, the goal is to reduce the given algebraic expression into its simplest form. This means we eliminate any unnecessary terms or factors. A common approach involves the use of exponent rules. This helps in rewriting the expression in a more manageable form without changing its value. In the exercise provided, simplifying the fraction was our first step. We used rules for dealing with exponents to combine like terms and simplify. This resulted in having a simpler expression before moving onto multiplication steps.

Key concepts to keep in mind when simplifying include:

• Combining like terms, such as terms with the same base and exponent.
• Reducing fractions to their simplest form.
• Carefully applying exponent rules to avoid errors.

Simplifying makes the problem easier to solve and often helps in seeing relationships between components of the expression.

Multiplication of Exponents

The multiplication of exponents involves applying the rule that states when you multiply two exponents with the same base, you can add their powers. This is represented by the formula:\[ a^m \cdot a^n = a^{m+n} \]
This rule allows us to simplify products of exponents quickly. In the exercise, after simplifying the fraction, we multiplied the resulting expression by the initial term. We added the exponents of the same base to simplify further.

To better understand this concept, let's consider the steps:

• First, identify terms with the same base. These are the ones that can be combined using this rule.
• Add the exponents for each base as you multiply the terms.

This principle is a cornerstone in working with exponential expressions, making problems faster and simpler to solve.

Division of Exponents

Division of exponents typically involves using the rule that states when you divide two exponents with the same base, you subtract the power of the denominator from the power of the numerator:\[ \frac{a^m}{a^n} = a^{m-n} \]
In our exercise, division was an integral step in simplifying the fraction \( \frac{x^4 y^8}{x^3 y^4} \). By using this rule, we simplified this fraction to \( xy^4 \), thus resulting in a simpler expression for further operations.

When performing division of exponents, remember to:

• Ensure the bases are identical before applying the exponent rules.
• Subtract the exponent in the denominator from the exponent in the numerator.
• Simplify the expression gradually, checking your work at each step to avoid mistakes.

Understanding and using this rule effectively saves time and effort when working with complex expressions.